and \(\Delta = \gamma c {\bf D}({\bf I}+\gamma c {\bf D})^{-1}\) .
In [1]}, it is observed that the integral above is not an instance
of the Harish-Chandra-Itzykson-Zuber (HCIZ) integral [2]} since the
\(n\times b\)  matrix \(\Phi \)  is not a square matrix.
In order to circumvent this problem, one has to extend matrix \(\Phi \mathsf {^H}\)  to the unitary \(b \times b\)  Haar matrix \(\widetilde{\Phi }\mathsf {^H}= [\Phi \mathsf {^H},\Phi _{\perp }\mathsf {^H}]\) , where \(\Phi _{\perp }\mathsf {^H}\)  is the orthogonal complement of \(\Phi \mathsf {^H}\)  with
respect to the unitary group \({\cal U}(b)\) . Thus, following [1]}, we can write
\( A =\frac{1}{|{\cal S}(b,n)||{\cal U}(b\mathord {-}n)|}\int \displaylimits _{{\cal U}(b)}{\rm e}^{\mathsf {Tr}\lbrace \Delta \Phi {\bf Y}\mathsf {^H}{\bf Y}\Phi \mathsf {^H}\rbrace } {\rm \,d}\widetilde{\Phi } \,. \) 
